I have a question about the phase transitions in the Ising model in the presence of a (constant) external magnetic field. I will state the question on $\mathbb Z^2$ for simplicity. A definition of the model is here.

It seems to be well known in the mathematical physics world that in the case of constant non-zero external magnetic field ($h_j= h>0\text{ for all $j\in\mathbb Z^2$ and }\mu>0$ in the notation of the Wikipedia article), there is no phase transition. That is, for all inverse temperatures $\beta$, the thermodynamic limits of the finite volume Gibbs measures with the all $+1$ boundary conditions and the all $-1$ boundary conditions are equal.

As far as I can tell, proofs of this are based on the Lee-Yang theorem (Messager, Miracle-Sole and Pfister) or on the GHS (Griffiths Hurst and Sherman) inequality. They are therefore based on the analytic properties of thermodynamic functions obtained by taking the limit over increasing domains.

Are there any elementary combinatorial proofs known? While hard to define exactly what I mean, what I have in mind is proofs based on estimating probabilities of configurations such as the proof of the existence of phase transition for Ising model in the absence of external field.